cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A307209 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).

Original entry on oeis.org

3, 5, 0, 4, 7, 8, 2, 9, 9, 9, 3, 3, 9, 7, 2, 8, 3, 7, 5, 8, 9, 1, 1, 2, 0, 5, 7, 0, 4, 3, 8, 0, 6, 1, 2, 5, 5, 8, 3, 8, 9, 3, 2, 4, 7, 8, 6, 2, 7, 1, 2, 7, 5, 3, 5, 4, 1, 9, 9, 4, 6, 2, 6, 6, 1, 4, 0, 5, 8, 3, 8, 5, 0, 3, 5, 0, 3, 4, 7, 5, 6, 3, 5, 2, 7, 4, 7, 5, 0, 9, 5, 0, 5, 1, 3, 7, 8, 9, 1, 7, 8, 4, 5, 9, 7
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 28 2019

Keywords

Comments

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.
A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...
Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).

Examples

			3.50478299933972837589112057043806125583893247862712753541994626614058385...

Links

Crossrefs

Cf. A073017, A307210, A307215, A324403, A324426, A324443.

Programs

  • Mathematica
    (* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^3 + j^3), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]
  • PARI
    default(realprecision, 50); exp(sumalt(k=1, -(-1)^k/k*sumnum(i=1, sumnum(j=1, 1/(i^3+j^3)^k)))) \\ 15 decimals correct

Formula

Equals limit_{n->infinity} A307210(n) / A324426(n).

A367956 a(n) = Product_{i=1..n, j=1..n} (i + 3*j).

Original entry on oeis.org

1, 4, 1120, 79833600, 3173289799680000, 123650071173117090201600000, 7337799401269093351612002462597120000000, 951792703318385182295191545713146608287219712000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 06 2023

Keywords

Crossrefs

Cf. A074962, A079478, A324402, A367957, A367958.

Programs

  • Mathematica
    Table[Product[i + 3*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

Formula

a(n) ~ A^(1/3) * 2^(16*n*(n+1)/3 + 13/18) * n^(n^2 - 19/36) / (Pi^(1/3) * Gamma(1/3)^(1/3) * 3^(n*(3*n+4)/2 + 11/36) * exp(3*n^2/2 + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant.

A367957 a(n) = Product_{i=1..n, j=1..n} (i + 4*j).

Original entry on oeis.org

1, 5, 2700, 567567000, 101370917007360000, 26995322179162164731904000000, 16635639072295355604762223305031680000000000, 34026881962001914598329145027742925521204742717440000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 06 2023

Keywords

Crossrefs

Cf. A074962, A079478, A324402, A367956, A367958.

Programs

  • Mathematica
    Table[Product[i + 4*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

Formula

a(n) ~ A^(1/4) * 5^(25*n*(n+1)/8 + 29/48) * n^(n^2 - 29/48) / (Pi^(1/4) * Gamma(1/4)^(1/2) * 2^(n*(4*n+5) + 5/6) * exp(3*n^2/2 + 1/48)), where A = A074962 is the Glaisher-Kinkelin constant.

A368064 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*i*j + j^2).

Original entry on oeis.org

1, 6, 24336, 870746557824, 1311726482483997806493696, 256433546267136937832915286844640487014400, 15678550451426175377500759401206644047210595564950427820202393600
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A367543, A324403, A367542, A079478^2, A368067, A368065, A368066.

Programs

  • Mathematica
    Table[Product[i^2 + 4*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 2^((3+sqrt(3))*n*(n+1) + (sqrt(3)-1)/6) * 3^(3*n*(n+1) + 13/24) * n^(2*n^2 - 7/6) / (Gamma(1/3)^(1/2) * Gamma(1/4)^(1/3) * Pi^(7/12) * (1 + sqrt(3))^((6*n*(n+1) + 1)/sqrt(3) - 1/2) * exp(3*n^2)).

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*i*j + j^2).

Original entry on oeis.org

1, 5, 12100, 188898484500, 91554454518735288960000, 4263420404009649597344435073399120000000, 46073465749493255153019723901007197815549903333795840000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A367543, A324403, A367542, A079478^2, A368064, A368065, A368066.

Programs

  • Mathematica
    Table[Product[i^2 + 3*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 5^(5*n*(n+1)/2 + 1/2) * n^(2*n^2 - 1) / (2 * Pi * exp(3*n^2) * phi^(sqrt(5)*(n*(n+1) + 1/6) - 1/2)), where phi = A001622 is the golden ratio.

A368685 a(n) = Product_{j=1..n, k=1..n} (j + k + n).

Original entry on oeis.org

1, 3, 600, 35562240, 1434015830016000, 70448433354492434841600000, 6610702315560389323908439364075520000000, 1709479709147705756603303596364188306401499545600000000000, 1660017838341811463102474357555838707949172571314554168163386261504000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2024

Keywords

Crossrefs

Cf. A079478, A306594, A324444, A368622, A368686.

Programs

  • Mathematica
    Table[Product[i+j+n, {i, 1, n}, {j, 1, n}], {n, 0, 8}]
Join[{1}, Table[3*BarnesG[n] * BarnesG[3*n] * Gamma[n]^2 * Gamma[3*n]^2 / (4*BarnesG[2*n]^2 * Gamma[2*n]^4), {n, 1, 8}]]

Formula

For n>0, a(n) = 3*BarnesG(n) * BarnesG(3*n) * Gamma(n)^2 * Gamma(3*n)^2 / (4*BarnesG(2*n)^2 * Gamma(2*n)^4).
a(n) ~ 3^(9*n^2/2 + 3*n + 5/12) * n^(n^2) / (2^(4*n^2 + 4*n + 5/6) * exp(3*n^2/2)).

A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

Original entry on oeis.org

1, 1, 6, 1440, 36288000, 184354652160000, 309071606732292096000000, 254046582743105184577722777600000000, 142518177743863255019484504453155074867200000000000
Offset: 0

Views

Author

Henry Bottomley, May 14 2005

Keywords

Examples

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*3!*1!*2!) = 34560/24 = 1440.

Crossrefs

Cf. A000178, A001700, A005130, A009963, A110131, A112332, A107254, A000142.

Programs

  • Magma
    [1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // G. C. Greubel, May 21 2019
  • Mathematica
    Table[Product[(n+k)!/(k+1)!,{k,0,n-1}],{n,0,10}] (* Alexander Adamchuk, Jul 10 2006 *)
a[n_] := (BarnesG[1 + 2 n] n! ((BarnesG[2 + n] Gamma[2 + n])/ BarnesG[3 + n])^(-1 + n))/BarnesG[2 + n]^2; Table[a[n], {n, 0, 10}] (* Peter Luschny, May 20 2019 *)
  • PARI
    {a(n) = prod(k=0,n-1, (n+k)!/(k+1)!)}; \\ G. C. Greubel, May 21 2019
  • Sage
    [product(factorial(n+k)/factorial(k+1) for k in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 21 2019

Formula

a(n) = (n+1)!*(n+2)!*...*(2n-1)!/(1!*2!*...*(n-1)!).
a(n) = A000178(2n-1)/(A000178(n)*A000178(n-1)).
a(n) = A079478(n)/A001813(n).
a(n) = A079478(n-1)*A006963(n+1).
a(n) = A107251(n)/A000108(n).
a(n) = A107251(n-1)*A009445(n-1).
a(n) = A107254(n)/A000142(n).
a(n) = A009963(2n-1, n-1).
a(n) = A009963(2n-1, n).
a(n) = (G(1+2*n)*n!*((G(2+n)*Gamma(2+n))/G(3+n))^(n-1))/G(2+n)^2, where G(x) is the Barnes G function. - Peter Luschny, May 20 2019
a(n) ~ A * 2^(2*n^2 - 7/12) * n^(n^2 - n - 5/12) / (sqrt(Pi) * exp(3*n^2/2 - n + 1/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 21 2019

A307210 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3 + 1).

Original entry on oeis.org

1, 3, 5100, 305727048000, 7748770873210669158912000, 476007332700693200670745550306381336371200000, 272661655519533773844144991586798737775635133552905539740860416000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 28 2019

Keywords

Comments

Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).
Product_{i=1..n, j=1..n} (1 + 1/(i^2 + j^2)) = A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313315993144237973600...

Crossrefs

Cf. A307209, A324403, A324426, A324443, A324444.

Programs

  • Maple
    a:= n-> mul(mul(i^3+j^3+1, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[i^3 + j^3 + 1, {i, 1, n}, {j, 1, n}], {n, 1, 8}]

Formula

a(n) ~ A307209 * A324426(n).
a(n) ~ c * A * 2^(2*n*(n+1) + 1/4) * exp(Pi*(n*(n+1) + 1/6)/sqrt(3) - 9*n^2/2 - 1/12) * n^(3*n^2 - 3/4) / (3^(5/6) * Pi^(1/6) * Gamma(2/3)^2), where c = A307209 = Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = 3.504782999339728375891120570... and A is the Glaisher-Kinkelin constant A074962.

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A307215 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^4 + j^4)).

Original entry on oeis.org

1, 9, 4, 0, 7, 3, 0, 2, 8, 5, 3, 7, 2, 3, 6, 1, 5, 2, 9, 9, 5, 3, 8, 6, 0, 7, 7, 5, 9, 9, 6, 4, 7, 7, 7, 2, 0, 3, 8, 7, 0, 7, 9, 6, 8, 2, 9, 3, 2, 1, 7, 0, 9, 2, 8, 1, 3, 0, 6, 1, 3, 9, 7, 4, 7, 2, 5, 2, 2, 6, 4, 2, 1, 7, 2, 0, 7, 2, 8, 3, 4, 7, 5, 5, 8, 9, 5, 3, 1, 0, 6, 8, 7, 6, 7, 7, 0, 7, 0, 0, 5, 9, 6, 1, 4
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 29 2019

Keywords

Comments

Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).
Product_{i=1..n, j=1..n} (1 + 1/(i^2 + j^2)) = A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313315993144237973600...
Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = A307209 = 3.50478299933972837589112...

Examples

			1.94073028537236152995386077599647772038707968293217092813061397472522642172...

Crossrefs

Cf. A307209, A324437.

Programs

  • Mathematica
    (* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^4 + j^4), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]

Formula

Equals limit_{n->infinity} (Product_{i=1..n, j=1..n} (1 + i^4 + j^4)) / A324437(n).

A324427 a(n) = Product_{k=1..n} (Product_{j=1..k} (Product_{i=1..j} (i+j+k))).

Original entry on oeis.org

1, 3, 360, 38102400, 109506663383040000, 337878174593229551661219840000000, 54048023654871725380569225530796717972337459200000000000, 25571582464158460440549345359703385621119611033206432205259362823202406400000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2019

Keywords

Crossrefs

Cf. A079478, A112332, A306594.

Programs

  • Maple
    a:= n-> mul(mul(mul(i+j+k, i=1..j), j=1..k), k=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[Product[Product[i+j+k, {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 0, 10}]
Table[Sqrt[Product[2^k Gamma[1 + 3*k/2]/Gamma[1 + k/2] (BarnesG[2 + k] BarnesG[2 + 3 k] )/BarnesG[2 + 2 k]^2 , {k, 1, n}]], {n, 0, 10}]
  • PARI
    a(n) = prod(k=1, n, prod(j=1, k, prod(i=1, j, i+j+k))); \\ Michel Marcus, Feb 27 2019

Formula

a(n) ~ 3^(3*n^3/4 + 9*n^2/4 + 47*n/24 + 7/24) * n^(n^3/6 + n^2/2 + n/3) / (2^(2*n^3/3 + 2*n^2 + 7*n/4 + 7/24) * exp(11*n^3/36 + 3*n^2/4 + n/3 - zeta(3)/(48*Pi^2))). - Vaclav Kotesovec, Nov 27 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023
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